How many 5 digit numbers are there, whose i digit is divisible by i?

Question

For example in the number 34567, the second digit (4) can be divided by 2, but the third digit (5) can't be divided by 3.

I thought that the first digit can be divided by any number as 1,2,3,4,5,6,7,8,9 are all divisible by 1.

The second digit can only be divided by 2, which is 2,4,6,8.

The third digit can only be divided by 3, which is 3,6,9.

The fourth digit can only be divided by 4, which is 4 or 8.

The fifth digit can only be divided by 5.

Answer 1

For the first position we have 9 choices 1,2,3,.....,9 (we cannot include 0 as we want 5 digit number).

For the second postion we have 5 choices 0,2,4,6,8 (here we don’t have problem in including 0).

Similarly for third, forth and fifth position we have 4, 3 and 2 choices respectively.

Now by multiplication rule in combinatorics (no need to take factorial)

We have total 9*5*4*3*2=1080 such numbers.

Answer to the second question is correct.

Answer 2

There are two mistakes:

1. $0$ is divisible by each number, therefore it can be selected on every position (except the first one, because then we wouldn't get 5-digit number)
2. There is no point in using permutations.